Perfectly Matched Layer

Transcription

1 7/2/217 Intucto D. Ramond Rumpf (915) Computational lectomagnetic Lectue #9 Pefectl Matched Lae Lectue 9Thee note ma contain copighted mateial obtained unde fai ue ule. Ditibution of thee mateial i tictl pohibited Slide 1 Outline Backgound Infomation The Uniaial Pefectl Matched Lae (UPML) Incopoating a UPML into Mawell quation Implementing the UPML Stetched Coodinate PML (SC PML) PML Pefomance UPML v SC PML Lectue 9 Slide 2 1

2 7/2/217 Backgound Infomation Lectue 9 Slide 3 Teno Teno ae a genealiation of a caling facto whee the diection of a vecto can be alteed in addition to it magnitude. Scala Relation V av V av Teno Relation a a a a V a a a V V a a a V Lectue 9 Slide 4 2

3 7/2/217 Reflectance fom a Suface with Lo Comple Refactive Inde n n j n odina efactive inde ocillation etinction coefficient deca Reflectance fom a lo uface ai R 1n 1n ** Lo contibute to eflection n n j Lectue 9 Slide 5 Reflection, Tanmiion and Refaction at an Inteface: Iotopic Cae Angle inc ef 1 n in n in Snell Law n, 1 1 n, 2 2 T Polaiation 2co11co2 T co co t T co1 co co n efactive inde in egion i i impedance in egion i TM Polaiation Lectue 9 i Slide 6 t TM TM 2co2 1co1 co co co1 co co

4 7/2/217 Mawell quation in Aniotopic Media Mawell cul equation in aniotopic media ae: H j j H Thee can alo be witten in a mati fom that make the teno apect of and moe obviou. H H j H H j H H Lectue 9 Slide 7 Tpe of Aniotopic Media Thee ae thee baic tpe of aniotopic media: io io io iotopic o o e uniaial a b c biaial Note: tem onl aie in the offdiagonal poition when the teno i otated elative to the coodinate tem. Lectue 9 Slide 8 4

5 7/2/217 (, ) V. (, ) Thee ae two wa to incopoate lo into Mawell equation. At ve low fequencie and/o fo time domain anali, the (, ) tem i uuall pefeed. H J jd j j At high fequencie and in the fequenc domain, (, ) i uuall pefeed. H jd j The paamete ae elated though j We ue thi fo FDTD Note: It doe not make ene to have a comple and a conductivit. Lectue 13 Slide 9 Mawell quation in Doubl Diagonall Aniotopic Media Mawell equation fo diagonall aniotopic media can be witten a H H H j j H H H We can genealie futhe b incopoating lo. H j H j j H j H j H H j j H H j H Lectue 9 Slide 1 5

6 7/2/217 Scatteing at a Doubl Aniotopic Inteface Refaction into a diagonall aniotopic mateial i decibed b Reflection fom a diagonall aniotopic mateial i in T TM bc in 1 2 aco aco aco aco bco bco bco bco a b c Sack, Zacha S., et al. "A pefectl matched aniotopic abobe fo ue a an abobing bounda condition." I Tan. Antenna and Popagation, Vol. 43, No. 12, pp , Lectue 9 Slide 11 Note on a Single Inteface It i a change in impedance that caue eflection Snell Law quantifie the angle of tanmiion Angle of tanmiion and eflection doe not depend on polaiation The Fenel equation quantif the amount of eflection and tanmiion Amount of eflection and tanmiion depend on the polaiation Lectue 9 Slide 12 6

7 7/2/217 Uniaial Pefectl Matched Lae (UPML) S. Zacha, D. Kingland, R. Lee, J. Lee, A Pefectl Matched Aniotopic Abobe fo Ue a an Abobing Bounda Condition, I Tan. on Ant. and Pop., Vol. 43, No. 12, pp , Lectue 9 Slide 13 Bounda Condition Poblem If we model a wave hitting ome device o object, it will catte the applied wave into potentiall man diection. We do NOT want thee catteed wave to eflect fom the boundaie of the gid. We alo don t want them to eente fom the othe ide of the gid (peiodic boundaie).?? How do we pevent thi? Lectue 9 Slide 14 7

8 7/2/217 How We Pevent Reflection in Lab In the lab, we ue anechoic foam to abob outgoing wave. Lectue 9 Slide 15 Abobing Bounda Condition We can intoduce lo at the boundaie of the gid! Abobing Bounda Lectue 9 Slide 16 8

9 7/2/217 Oop!! But if we intoduce lo, we alo intoduce eflection fom the lo egion!! R 1n 1n Lectue 9 Slide 17 Match the Impedance We need to intoduce lo to abob outgoing wave, but we alo need to match the impedance to the poblem pace to pevent eflection. intoduce lo hee j adjut thi to contol impedance Lectue 9 Slide 18 9

10 7/2/217 Moe Touble? B eamining the Fenel equation, we ee that we can onl pevent eflection fom the inteface at one fequenc, one angle of incident, and one polaiation. co co co T 2 1 2co11co2 co1 co co co TM 2 1 1co12co2 co2 Lectue 9 Slide 19 Aniotop to the Recue!! It tun out we can pevent eflection at all angle and fo all polaiation if we allow ou abobing mateial to be doubldiagonall aniotopic. and and Lectue 9 Slide 2 1

11 7/2/217 Poblem Statement fo the PML Fee Space, % 1 2 1, 1% Lectue 9 Slide 21 Deigning Aniotop fo Zeo Reflection (1 of 3) We need to pefectl match the impedance of the gid to the impedance of the abobing egion. evewhee One ea wa to enue impedance i pefectl matched i: a b c Lectue 9 Slide 22 11

12 7/2/217 Deigning Aniotop fo Zeo Reflection (2 of 3) If we chooe bc 1, then the efaction equation educe to in bc in in No efaction! The eflection coefficient now educe to T TM aco1 bco2 a b aco bco a b 1 2 aco1 bco2 a b aco bco a b 1 2 Thee ae no longe a function of angle!! Lectue 9 Slide 23 Deigning Aniotop fo Zeo Reflection (3 of 3) If we futhe chooe a b, the eflection equation educe to T TM a b a b a b a b Reflection will alwa be eo egadle of fequenc, angle of incidence, o polaiation!! Recall the necea condition: bc 1 and a b Lectue 9 Slide 24 12

13 7/2/217 The PML Paamete (1 of 3) So fa, we have a b c 1 ab c Thu, we can wite ou PML in tem of jut one paamete. S j 1 Thi fom of teno i wh we call thi a uniaial PML. Thi i fo a wave tavelling in the + diection incident on a ai bounda. Lectue 9 Slide 25 The PML Paamete (2 of 3) We potentiall want a PML along all the bode. 1 S S S 1 1 Thee can be combined into a ingle teno quantit. S S S S Lectue 9 Slide 26 13

14 7/2/217 The PML Paamete (3 of 3) The 3D PML can be viualied thi wa S Lectue 9 Slide 27 UPML in Clindical and Spheical Coodinate S Clindical Coodinate S Spheical Coodinate 2 1 F. L. Teieia, W. C. Chew, Stematic Deivation of Aniotopic PML Abobing Media in Clindical and Spheical Coodiante, I Micowave and Guided Wave Lette, Vol. 7, No. 11, pp , Lectue 9 Slide 28 14

15 7/2/217 Two Dimenional UPML Fo 2D imulation in the plane, = 1 and the UPML teno educe to S Lectue 9 Slide 29 Incopoating a UPML into Mawell quation Lectue 9 Slide 3 15

16 7/2/217 Incopoating the UPML Into Mawell q. Mawell quation Thi et of equation doe include device, but no UPML at the bounda to abob outgoing wave. k H H k UPML Thi et of equation include the UPML to abob outgoing wave, but doe not include device o eal mateial. k H H k S S Mawell quation with UPML k H H k S S Thi appoach incopoate the PML in a wa that i independent of the mateial. It keep the PML impedance matched to the backgound mateial automaticall. Lectue 9 Slide 31 Mawell quation with a UPML Mawell equation with a UPML k SH H k S The UPML can be incopoated into the mateial teno diectl. k H S H k S Lectue 9 Slide 32 S Thi let u fomulate and implement a numeical algoithm without having to eplicitl conide the PML. It i impl incopoated into the mateial teno. 16

17 7/2/217 Vecto panion Auming onl diagonal teno Mawell equation epand to k k k H H H H H k H H k H H k Lectue 9 Slide 33 Abob UPML into and (3D Gid) We can abob the UPML paamete into the mateial function. We can now wite Mawell equation a k H kh k H H H H H H H k k k Thi mean we can fomulate a code a if thee wa no PML. All we have to do i modif the mateial being modeled nea the boundaie. Lectue 9 Slide 34 17

18 7/2/217 Abob UPML into and (2D Gid) Let be the unifom diection, then d/d = and = 1. We can till abob the UPML paamete into the mateial function. We can now wite Mawell equation a Mode H Mode H H k k H k H H k kh H k Lectue 9 Slide 35 Implementing the PML Lectue 9 Slide 36 18

19 7/2/217 The Pefectl Matched Lae (PML) The pefectl matched lae (PML) i an abobing bounda condition (ABC) whee the impedance i pefectl matched to the poblem pace. Reflection enteing the lo egion ae pevented becaue impedance i matched. Reflection fom the gid bounda ae pevented becaue the outgoing wave ae abobed. PML PML Poblem Space PML PML Lectue 9 Slide 37 Tpical Gid Scheme 2 cell 2 cell PML 2 cell PML pace egion PML Peiodic Bounda Peiodic Bounda PML PML Peiodic Device 2 cell Finite Device Lectue 9 Slide 38 PML 19

20 7/2/217 Jutification fo the Space Region The efactive inde i high inide the PML o evanecent wave can become popagating wave, giving an ecape path fo powe. PML PML PML PML Lectue 9 Slide 39 How to Calculate the PML Paamete Mawell q. with PML k H H k NGRID = [N N]; NPML = [ 2 2]; [,] = calcpml2d(ngrid,npml); 1 a j 1 1 a j a j p 1 ma p 1 ma p 1 a a L a a L a a L Computing PML Paamete fee pace impedance Lectue 9 Slide 4 ma ama 5 3 p 5 1 ma 2 ma in 2L 2 ma in 2L 2 ma in 2L Witing thi function will be in homewok 2

21 7/2/217 Viualiing the PML Lo Tem 2D Fo bet pefomance, the lo tem hould inceae gaduall into the PML. Lectue 9 Slide 41 Pocedue fo Calculating and on a 2D Gid 1. Initialie and to all one.,, 1 2. Fill in ai PML egion uing two fo loop. NXLO NXHI 3. Fill in ai PML egion uing two fo loop. NYLO NYHI Lectue 9 Slide 42 21

22 7/2/217 Note About /L, /L, and /L The following atio povide a ingle quantit that goe fom to 1 a ou move though a PML egion. and and L L L,, poition within PML L, L, L ie of PML We can calculate the ame atio uing intege indice fom ou gid. n n o L NXLO NXHI n n o L NYLO NYHI n o L NZLO n NZHI n = 1, 2,..., NXLO n = 1, 2,..., NXHI n = 1, 2,..., NYLO n = 1, 2,..., NYHI n = 1, 2,..., NZLO n = 1, 2,..., NZHI 1 L 1 L Lectue 9 Slide 43 Viualiing in 2D % ADD XLO PML fo n = 1 : NXLO (NXLO-n+1,:) =... end () = a ()[1+j ()] () = 1 () = a ()[1+j ()] % ADD XHI PML fo n = 1 : NXHI (N-NXHI+n,:) =... end 1 NXLO Lectue 9 Slide 44 N-NXHI+1 N 22

23 7/2/217 Viualiing in 2D 1 NYLO () = a ()[1+j ()] % ADD YLO PML fo n = 1 : NYLO (:,NYLO-n+1) =... end () = 1 N NYHI + 1 N () = a ()[1+j ()] % ADD YHI PML fo n = 1 : NYHI (:,N-NYHI+n) = end Lectue 9 Slide 45 ample Data fo 2D NGRID = [7 4]; NPML = [ ]; [,] = calcpml2d(ngrid,npml); a ma ma 3 p 3 1 = 1.e+3 * i i i i i i i i i i i i i i i i i i i i = 1.e+3 * i i i i i i i i i i i i i i i i i i i i i Lectue 9 Slide 46 23

24 7/2/217 PML i Not a Bounda Condition A numeical bounda condition i the ule ou follow when an equation efeence a field fom outide the gid. The PML doe not adde thi iue. It i impl a wa of incopoating lo while peventing eflection o a to abob outgoing wave. Sometime it i called an abobing bounda condition, but thi i till mileading a the PML i not a tue bounda condition. Lectue 9 Slide 47 Stetched Coodinate Pefectl Matched Lae (SC PML) Lectue 9 Slide 48 24

25 7/2/217 The Uniaial PML Mawell equation with uniaial PML ae: k S S H H k S Lectue 9 Slide 49 Reaange the Tem We can bing the PML teno to the left ide of the equation and aociate it with the cul opeato. 1 k H S H k 1 S The cul opeato i now S Lectue 9 Slide 5 25

26 7/2/217 Stetched Coodinate Ou new cul opeato i S The facto,, and ae effectivel tetching the coodinate, but the ae tetching into a comple pace. Lectue 9 Slide 51 Dop the Othe Tem We dop the non tetching tem Jutification 1 1 Inide the PML, = = 1. Thi i valid evewhee ecept at the eteme cone of the gid whee the PML ovelap. Thi alo implie that the UPML and SC PML have neal identical pefomance in tem of eflection, enitivit to angle of incidence, polaiation, etc. Lectue 9 Slide 52 26

27 7/2/217 Mawell quation with a SC PML Mawell equation befoe the PML i added ae k H H k The SC PML i incopoated a follow. j H H j Lectue 9 Slide 53 Vecto panion Mawell equation with a SC PML epand to Full Aniotopic 1 H 1 H k 1 H 1 H k 1 H 1 H k 1 1 k H H H 1 1 k H H H 1 1 k H H H Diagonall Aniotopic 1 H 1 H k 1 H 1 H k 1 H 1 H k k H 1 1 kh k H Lectue 9 Slide 54 27

28 7/2/217 PML Pefomance Lectue 9 Slide 55 PML Ae Not Pefect PML abobing bounda condition ae not pefect abobe. The till eflecte wave! Lectue 9 Slide 56 28

29 7/2/217 Theoetical Pefomance Given the following choice of PML paamete aˆ aˆ aˆ Lectue 9 Slide 57 1 j,ma L 1 j,ma L m 1 j,ma L We chooe i,ma to achieve a taget maimum eflectance R at nomal incidence accoding to i,ma m1lnr 2 L i We tpicall chooe 3m 4 4 i,ma i m m UPML Pefomance in FDFD ama 3 UPML pefomance if affected b ] and it ie. p 3 H mode UPML ehibit lightl pooe pefomance. ma 1 Lectue 9 Slide 58 29

30 7/2/217 UPML V. SC PML Lectue 9 Slide 59 UPML V. SC PML Benefit Uniaial PML Ha a phical intepetation Model can be fomulated and implemented without conideing the PML in the fequenc domain Stetched Coodinate PML Benefit Le computationall intenive in time domain Moe efficient implementation in the time domain Matice ae bette conditioned. Dawback Can be moe computationall intenive to implement in timedomain Reulting matice ae le well conditioned in the fequendomain Dawback Mut be accounted fo in the fomulation and implementation of the numeical method. Not intuitive to undetand Lectue 9 Slide 6 3